Inspired by a previous post, Embed a Spin group to a special unitary group
I am wondering what are the magic properties of the quotient space from $$U(2^{l-1})/{\rm Spin}(2 l)$$ that makes such an embedding possible: $${\rm Spin}(2 l) \subset U(2^{l-1}).$$
For example, I know that ${\rm Spin}(k+1)/{\rm Spin}(k)=S^k$ is a $k$-sphere.
Do we have similar properties to describe the quotient space from $U(2^{l-1})/{\rm Spin}(2 l)$?
What are the homotopy groups of this space? $\pi_d(U(2^{l-1})/ {\rm Spin}(2 l))=?$
Any references are welcome.